People's decisions depend on the way probability is expressed to them. A lot of financial models assume Normal / Gaussian distribution due to its mathematical tractability. However, as a thin-tailed model, it underestimates the likelihood of extreme events when compared to many real-world phenomena. On the other hand, other distributions, like Student's t or Pareto, can account for extreme values with their fat / heavy tails, which is relevant in the context of modelling the occurrence of outlier events. This property makes it more appropriate for financial returns data (especially at higher frequencies like daily returns), where it's important to estimate the likelihood of very large losses. The difference between the Normal and the other mentioned distributions can be quantified as "unexpectedness", or "surprise".
| what people assume / under normal circumstances | what actually happens / tools of comparison |
|---|---|
| Gaussian | Pareto / Cauchy / Studen t |
| lower bounds, less variation, stable / finite std | use mean absolute deviation (MAD) instead (less sensitive to extreme values) |
| a suspected average | no actual average |
| outliers: single event is not expected to have a big effect | outliers: single event has more effect than we expect. |
| the speed of effect is expected to be more immediate | the speed of effect: slower than we think |
Standard deviation is a more conservative measure of dispersion for data with outliers, since it tends to give a larger spread. However, MAD can give a better idea of the "typical" deviation, since it's not as influenced by extreme values.
Tesla returns: 1.64% of data is beyond 3 standard deviations
Let's try to quantify how much a particular price movement deviates from what we would expect under normal circumstances. Assuming TSLA's daily returns follow a fat-tailed distribution, most days will see relatively small price changes, but occasionally, there will be very large changes -- these are the "fat tails".
One way to quantify the "surprise" of a price movement is by calculating how many standard deviations away from the mean it is (or "z-score"). Under a normal distribution, a z-score of 3 or more is considered very rare (occurring less than 0.3% of the time), but under a fat-tailed distribution, such occurrences are more common.
Let's say, for example, the average daily return of TSLA is 0.002% with a standard deviation of 0.036%. If TSLA one day has a return of 0.24%, we can calculate the z-score as follows:
$z = (0.24\% - 0.002\%) / 0.036\% = 6.7$
This suggests the move is 6.7 standard deviations away from the mean, a huge "surprise" under normal distribution assumptions, but in a fat-tailed world, such moves can happen more often than we would normally expect.
Surprise can be measured as the difference between the earnings expectations (Gaussian), and actual earnings (nonlinear, based on a variety of factors including changes in market conditions, the company's strategic decisions, and unforeseen events).
Analysts' expectations are usually reflected in the form of earnings estimates, which are frequently collected and reported by a number of financial information and news services. These include:
Earnings Forecasts: These are typically expressed as an estimate of Earnings Per Share (EPS) for a given quarter or year. The EPS estimate represents the analysts' average expectation of a company's profitability.
Revenue Estimates: Analysts also provide estimates for a company's revenue for a given period. This helps investors understand the expected top-line growth of the company.
Other Financial Metrics: Depending on the company and industry, analysts may also provide estimates for other key financial metrics, such as gross margin, operating margin, EBITDA, and more.
We could look into approximating the concept of "priced in" - the extent to which market expectations, including all publicly available information, are already reflected in the current price of a security. Some of the factors that might be "priced in":
Publicly Available Information: This includes financial reports, news releases, and any other information that's publicly available and pertinent to the value of the asset.
Market Sentiment: This is the overall attitude of investors towards a particular security or financial market. It is the cumulative attitude of all market participants towards risk, and it can heavily influence whether potential news or events are already priced in.
Analyst Estimates: If analysts have widely shared an expectation for a company’s future performance, this expectation may be priced in.
Economic Indicators: Things like GDP reports, unemployment rates, and consumer sentiment indices can all impact what's priced into the market.